Right triangles are nice and neat, with their side lengths obeying
the Pythagorean Theorem. Any two right triangles with the same two
non-right angles are "similar", in the technical sense that their corresponding sides
are in proportion. For instance, the following two triangles (not drawn to scale) have all
the same angles, so they are similar, and the corresponding pairs of their sides are in proportion:

Around the fourth or fifth century AD, somebody very clever living
in or around India noticed these consistency of the proportionalities of right triangles with the
same sized base angles, and started working on tables of ratios corresponding to those non-right
angles. There would be one set of ratios for the one-degree angle in a 1-89-90 triangle, another
set of ratios for the two-degree angle in a 2-88-90 triangle, and so forth. These ratios are called
the "trigonometric" ratios for a right triangle.

Given a right triangle with a non-right angle designated as
θ ("THAY-tuh"), we can label the hypotenuse (always
the side opposite the right angle) and then label the other two sides "with respect
to θ"
(that is, in relation to the non-right angle θ
that we're working with).

The side opposite the angle θ
is the "opposite" side, and the other side, being "next" to the angle
(but not being the hypotenuse) is the "adjacent" side.

For the same triangle, if we called the third angle β ("BAY-tuh"),
the labelling would be as shown:

As you can see, the labels "opposite" and "adjacent"
are relative to the angle in question.

There are six ways to form ratios of the three sides of this triangle.
I'll shorten the names from "hypotenuse", "adjacent", and "opposite" to
"hyp", "adj", and "opp":

name

ratio

notation

opp/hyp

adj/hyp

opp/adj

name

ratio

notation

hyp/opp

hyp/adj

adj/opp

Each of these ratios has a name:

name

ratio

notation

sine

opp/hyp

cosine

adj/hyp

tangent

opp/adj

name

ratio

notation

cosecant

hyp/opp

secant

hyp/adj

cotangent

adj/opp

...and each of these names has an abbreviated notation, specifying
the angle you're working with:

name

ratio

notation

sine

opp/hyp

sin(θ)

cosine

adj/hyp

cos(θ)

tangent

opp/adj

tan(θ)

name

ratio

notation

cosecant

hyp/opp

csc(θ)

secant

hyp/adj

sec(θ)

cotangent

adj/opp

cot(θ)

The ratios in the left-hand table are the "regular" trig
ratios; the ones in the right-hand table are their reciprocals (that is, the inverted fractions).
To remember the ratios for the regular trig functions, many students use the mnemonic SOH CAH TOA,
pronounced "SOH-kuh-TOH-uh" (as though it's all one word). This mnemonic stands for:

Sine is Opposite over Hypotenuse

Cosine is Adjacent over Hypotenuse

Tangent is Opposite over Adjacent

When two ratios have the same name other than the "co-"
at the start of one of them, the pair are called "co-functions". (Note: The ratios didn't
get their current names until the 12th century or so, as a result of Europeans making some mistakes
when they translated Arabic texts.)